Tour:Subgroup containment relation equals divisibility relation on generators
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This article adapts material from the main article: subgroup containment relation in the group of integers equals divisibility relation on generators
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WHAT YOU NEED TO DO:
- Read and understand the main statement, as well as the corollaries.
- Convince yourself of the truth of the main statement, as well as of how the corollaries follow from it.
Statement
Let denote the Group of integers (?) under addition. Suppose
are subgroups of
. Suppose
and
. Then,
if and only if
, i.e.,
is a multiple of
.
Note that, because every nontrivial subgroup of the group of integers is cyclic on its smallest element, the subgroups and
can be written in the form
respectively.
Related facts
Corollaries
- Given two subgroups
and
, their intersection is the subgroup generated by an element
with the property that
, and if
, then
. Such an integer
is termed a least common multiple of
and
(if we allow only nonnegative integers, then it is unique).
- Given two subgroups
and
, their join is the subgroup generated by an element
with the property that
, and if
, then
. Such an integer
is termed a greatest common divisor of
and
(if we allow only nonnegative integers, then it is unique).
- The greatest common divisor of
and
can be written as
for some integers
and
. That is because it is in the subgroup generated by
and
.
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PREVIOUS: Every nontrivial subgroup of the group of integers is cyclic on its smallest element| UP: Introduction four (beginners)| NEXT: No proper nontrivial subgroup implies cyclic of prime order